11-5 Area or Circles and Sectors

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Warm Up

1. Find w, y, and z. Give the answers in simplest radical form.

11.5 Areas of Circles and Sectors

Warm Up Continued

Find each measure.

54 2. RT

3. UQ

11.5 Areas of Circles and Sectors

Find the area of a circle and a sectors of a circle.

Objectives

11.5 Areas of Circles and Sectors

sector of a circlesegment of a circle

Vocabulary

11.5 Areas of Circles and Sectors

You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram.

The base of the parallelogram is about half the circumference, or r, and the height is close to the radius r. So A r · r = r2.

The more pieces you divide the circle into, the more accurate the estimate will be.

11.5 Areas of Circles and Sectors

11.5 Areas of Circles and Sectors

11.5 Areas of Circles and Sectors

The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the

circle by

11.5 Areas of Circles and Sectors

11.5 Areas of Circles and Sectors

Write the degree symbol after m in the formula to help you remember to use degree measure not arc length.

Helpful Hint

11.5 Areas of Circles and Sectors

Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth.

Example 1A: Finding the Area of a Sector

sector HGJ

Use formula for area of sector.

Substitute 12 for r and 131 for m.

= 52.4 m2 164.62 m2 Simplify.

11.5 Areas of Circles and Sectors

Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth.

Example 1B: Finding the Area of a Sector

sector ABC

Use formula for area of sector.

Substitute 5 for r and 25 for m.

1.74 ft2 5.45 ft2 Simplify.

11.5 Areas of Circles and Sectors

Check It Out! Example 1a

Find the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.

sector ACB

Use formula for area of sector.

Substitute 1 for r and 90 for m.

= 0.25 m2 0.79 m2 Simplify.

11.5 Areas of Circles and Sectors

Check It Out! Example 1b

Find the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.

sector JKL

Use formula for area of sector.

Substitute 16 for r and 36 for m.

= 25.6 in2 80.42 in2 Simplify.

11.5 Areas of Circles and Sectors

A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122°?

Example 2: Automobile Application

Use formula for area of sector.

r = 18 in.

345 in2 Simplify.

11.5 Areas of Circles and Sectors

Check It Out! Example 2

To the nearest square foot, what is the area watered in Example 2 (p. 765) as the sprinkler rotates through a semicircle?

Use formula for area of sector.

d = 720 ft, r = 360 ft

203,575 ft2 Simplify.

11.5 Areas of Circles and Sectors

A segment of a circle is a region bounded by an arc and its chord.

11.5 Areas of Circles and Sectors

Find the area of segment LNM to the nearest hundredth.

Example 3: Finding the Area of a Segment

Use formula for area of sector.

Substitute 9 for r and 120 for m.

= 27 cm2 Simplify.

Step 1 Find the area of sector LNM.

11.5 Areas of Circles and Sectors

Find the area of segment LNM to the nearest hundredth.

Example 3 Continued

Simplify.

Step 2 Find the area of ∆LNM. Draw altitude NO.

11.5 Areas of Circles and Sectors

In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √3 times the length of the shorter leg.

Remember!

11.5 Areas of Circles and Sectors

Find the area of segment LNM to the nearest hundredth.

Example 3 Continued

Step 3

area of segment = area of sector LNM – area of ∆LNM

49.75 cm2

11.5 Areas of Circles and Sectors

Check It Out! Example 3

Find the area of segment RST to the nearest hundredth.

Use formula for area of sector.

Substitute 4 for r and 90 for m.

= 4 m2 Simplify.

Step 1 Find the area of sector RST.

11.5 Areas of Circles and Sectors

Simplify.

Step 2 Find the area of ∆RST.

Check It Out! Example 3 Continued

Find the area of segment RST to the nearest hundredth.

ST = 4 m, and RS = 4m.

= 8 m2

11.5 Areas of Circles and Sectors

Step 3

area of segment = area of sector RST – area of ∆RST

4.57 m2

Find the area of segment RST to the nearest hundredth.

Check It Out! Example 3 Continued

= 4 – 8

11.5 Areas of Circles and Sectors

In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle.

11.5 Areas of Circles and Sectors

Find each arc length. Give answers in terms of and rounded to the nearest hundredth.

Example 4A: Finding Arc Length

FG

Use formula for area of sector.

5.96 cm 18.71 cm

Substitute 8 for r and 134 for m.

Simplify.

11.5 Areas of Circles and Sectors

Find each arc length. Give answers in terms of and rounded to the nearest hundredth.

Example 4B: Finding Arc Length

an arc with measure 62 in a circle with radius 2 m

Use formula for area of sector.

0.69 m 2.16 m

Substitute 2 for r and 62 for m.

Simplify.

11.5 Areas of Circles and Sectors

Check It Out! Example 4a

Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

GH

Use formula for area of sector.

Substitute 6 for r and 40 for m.

Simplify.= m 4.19 m

11.5 Areas of Circles and Sectors

Check It Out! Example 4b

Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

an arc with measure 135° in a circle with radius 4 cm

Use formula for area of sector.

= 3 cm 9.42 cm

Substitute 4 for r and 135 for m.

Simplify.

11.5 Areas of Circles and Sectors

Lesson Quiz: Part I

Find each measure. Give answers in terms of and rounded to the nearest hundredth.

1. area of sector LQM

2.5 in. 7.85 in.

7.5 in2 23.56 in2

2. length of NP

11.5 Areas of Circles and Sectors

Lesson Quiz: Part II

3. The gear of a grandfather clock has a radius of 3 in. To the nearest tenth of an inch, what distance does the gear cover when it rotates through an angle of 88°?

4.6 in.

4. Find the area of segment GHJ to the nearest hundredth.

55.94 m2

11.5 Areas of Circles and Sectors